That’s the question Apple CEO Tim Cook has faced in the two years since his predecessor, revered Apple co-founder Steve Jobs, died after a long battle with cancer. Cook, a 52-year-old operational wizard from Robertsdale, Ala., took over as Apple CEO a few months before Jobs died on October 5th, 2011. His challenge has been to maintain the momentum at the world’s largest and most famous technology company, following a decade-long period of turn-around and innovation with few parallels in corporate history.

Over the last year, Apple’s once-high flying stock price has shed a quarter of its value, as Wall Street analysts and investors alike have begun to question whether the firm’s greatest days are in the past. Indeed, it’s hard to imagine any executive, no matter how talented, matching Jobs’ track record. Over his career, Jobs radically disrupted at least eight industries: personal computing, computer operating systems, music, mobile phones, publishing, tablet computers, and Hollywood animation. The iPod, the iPhone, and the iPad transformed or created their respective markets, and a first generation iPod is on display at the Museum of Modern Art in New York City. Jobs was named the CEO of the decade in 2009 and, for many, he was the CEO of the century.

Jobs, with Cook by his side as chief operating officer, turned Apple into a $440 billion juggernaut partly by developing a sophisticated supply chain—cornering the market for flash memory in the mid-2000s, for example, well before it became clear that technology would become a staple of devices ranging from phones to laptops—as well as by aggressively releasing glitzy new devices, even if they risked cannibalizing sales of best-selling products. As recent books like Inside Apple and Walter Isaacson’s biography, Steve Jobs, have shown, that’s how Jobs and company essentially created the world’s largest, most successful startup. The 1998-2011 period in Apple’s history is likely to become an object lesson in avoiding the so-called “innovator’s dilemma.”

At the heart of the discussion about Apple’s future now is a debate about radical innovation versus incrementalism. Apple investors, consumers, and analysts grew accustomed to watching the company unveil stunning new products with great fanfare. But since the introduction of the iPad three years ago, none of the breakthrough products—a television or smartwatch—the company is rumored to be working on have materialized. In a recent interview with Bloomberg BusinessWeek, Cook made a clear argument for incrementalism, but he couched that argument in terms of innovation. “Some people see innovation as change, but we have never really seen it like that,” Cook told the magazine. “It’s making things better.”

Then there’s the question of whether Apple’s advantages are starting to erode. Apple enjoyed a head start in the smart phone market with the iPhone and in tablets with the iPad. But there are clear signs that the company’s rivals are beginning to catch up. Samsung, the South Korean electronics titan, has surged to become the largest handset maker in the world. Google’s Android operating system has racked up massive global market-share and is now the world’s top mobile OS. Still, Apple accounted for 69% of the smart phone industry’s 2012 profits, according to Canaccord Genuity analyst T. Michael Walkley. Apple generates more than $1 billion per month from iPhone sales, which account for more than half of the company’s revenue.

Apple has a winning product in the iPad tablet, but that device faces emboldened competition, most notably from new Google, Samsung and Microsoft products. The iPad remains the most popular tablet, but the field is growing increasingly crowded. In the second quarter of the 2013, Apple shipped 14.6 million iPads, down from 19.5 million in in the first quarter, according to research firm IDC. Part of that decline was driven by the fact that Apple didn’t launch a new tablet in the first quarter, as it has in past years, but increasing competition also played a role. For example, second-place Samsung shipped 8.1 million units in the second quarter, a whopping 277% increase from 2.1 million units shipped in the same period in 2012. And in terms of tablet operating systems, Android is now the top platform, with 62.6% market share, compared to 32.5% for Apple’s iOS platform.

Naturally, Apple shareholders and Wall Street analysts focus on the company’s stock price, and over the last year, Apple has shed more than $100 billion in market capitalization. Cook doesn’t seem too worried about that, though he’s said that he’s not thrilled about his company’s stalling stock price. Although Cook has taken actions to stem the decline, including a robust stock buyback program, Apple shares are down 10% this year, even as the tech-heavy NASDAQ index has soared by more than 20% (as has Google’s stock price).

Apple’s Wall Street woes have attracted some sharks, most notably Carl Icahn, the 77-year-old activist investor who made his name as a corporate raider. Icahn, who is worth an estimated $20 billion, has a long and colorful history of taking large positions in companies that he believes to be undervalued and then agitating for change. Apple is Icahn’s latest target, and he’s called for the company to return more of its massive cash hoard to investors. Icahn wants Apple to take advantage of low interest rates to borrow a whopping $150 billion at 3% interest, to be used to buy back its own stock. On Monday, Icahn, who owns more than $2 billion worth of Apple stock, met with Cook in New York City to discuss the matter. The meeting was “cordial,” Icahn later tweeted, but he added in an interview with CNBC that things did get a little “testy.” Icahn told CNBC: “I can promise you that I’m not going away until they hear a lot more from me concerning this.”

Still, despite the stock slump, Apple’s core business continues to power ahead. The new iPhones are a huge hit. Last Monday, Apple said it sold a record nine million units of the latest versions during the first weekend they were on sale. In a now-familiar scene, consumers lined up for hours at Apple stores across the country. Cook clearly commands great respect on Wall Street and in Silicon Valley. Though he has yet to bring entirely new products to market, Cook has shown that he’s willing to shake up Apple’s management if he deems it necessary, including firing long-time mobile software head Scott Forstall and Apple Store chief John Browett, who simply wasn’t a good fit with the company’s corporate culture.

Cook doesn’t have to look far to see what can happen when a tech giant rests on its laurels. But the lingering question is whether incremental improvements to existing products will be enough to power Apple earnings over the next 5 years—venture capital titan Marc Andreessen’s baseline time horizon for technology innovation—or whether the company will need new breakthrough products.